I feel your pain! The classic question about what to measure, because that's what people optimize for. In your case though it looks like you are already measuring things other than just first responsi time and numbers of replies - and yet your Support Team ignores other metrics. And my first question is: why is that? What is the incentive to optimize one measurement over the other?
The incentive can be quite subtle. Do you praise support reps for high reply numbers, but don't ever mention IQS? Does your business overview always includes number of replies sent, but doesn't mention documents improved thanks to detailed notes left by support reps? Is volume something people boast about (because quantitative) and being really good at diffusing first angry contacts is something no one notices? How can you draw positive attention to metrics outside of the norm?
@Triin Ilves167 already mentioned the infographic where you can find all the things to track. While looking at the options, keep in mind that the deeper question about what type of behaviour do you want to encourage with your metrics. And then optimize for that - in tracking, but also in how you talk about and respond to metrics changing over time. And if you can tie those back into your support creed or company values - that's even more powerful.